Abstract. We consider circle packings in the hyperbolic plane,
by finitely many congruent circles, which maximize the number of
touching pairs. We show that such a packing has all of its centers
located on the vertices of a triangulation of the hyperbolic plane
by congruent equilateral triangles, provided the diameter d of the
circles is such that an equilateral triangle in the hyperbolic plane
of side length d has each of its angles is equal to 2 pi/N for some
N > 6.